Hereditary realizations of linear neutral delay differential systems

Abstract

The realization problem for hereditary linear neutral delay differential systems is considered in the context of systems over commutative rings. The notion of formal spectral minimality is introduced and related to the general notion of spectral minimality of infinite dimensional systems. It is shown that for neutral delay systems, unlike retarded delay systems, absolute minimality does not imply formal spectral minimality. For systems with commensurable delays it is shown that an absolutely minimal system may always be reduced to a formally spectrally mininal system by an extended type of system transformation which we call weak restricted system equivalence following the terminology of Rosenbrock. This result is used to give a two-step realization procedure for the construction of canonical neutral realizations for a large class of non-rational transfer functions. A counterexample is given to show that, in the case of non-commensurable delays, there may not exist an hereditary realization which is both absolutely minimal and (formally) spectrally minimal.